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Theorem psubclsubN 39468
Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclsub.s 𝑆 = (PSubSpβ€˜πΎ)
psubclsub.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
psubclsubN ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 ∈ 𝑆)

Proof of Theorem psubclsubN
StepHypRef Expression
1 eqid 2725 . . 3 (βŠ₯π‘ƒβ€˜πΎ) = (βŠ₯π‘ƒβ€˜πΎ)
2 psubclsub.c . . 3 𝐢 = (PSubClβ€˜πΎ)
31, 2psubcli2N 39467 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋)
4 eqid 2725 . . . . . . 7 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
54, 1, 2psubcliN 39466 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋))
65simpld 493 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
7 psubclsub.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
84, 7, 1polsubN 39435 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) ∈ 𝑆)
96, 8syldan 589 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) ∈ 𝑆)
104, 7psubssat 39282 . . . 4 ((𝐾 ∈ HL ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) ∈ 𝑆) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
119, 10syldan 589 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
124, 7, 1polsubN 39435 . . 3 ((𝐾 ∈ HL ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) ∈ 𝑆)
1311, 12syldan 589 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) ∈ 𝑆)
143, 13eqeltrrd 2826 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3940  β€˜cfv 6542  Atomscatm 38790  HLchlt 38877  PSubSpcpsubsp 39024  βŠ₯𝑃cpolN 39430  PSubClcpscN 39462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-proset 18284  df-poset 18302  df-lub 18335  df-glb 18336  df-join 18337  df-meet 18338  df-p1 18415  df-lat 18421  df-clat 18488  df-oposet 38703  df-ol 38705  df-oml 38706  df-ats 38794  df-atl 38825  df-cvlat 38849  df-hlat 38878  df-psubsp 39031  df-pmap 39032  df-polarityN 39431  df-psubclN 39463
This theorem is referenced by:  pclfinclN  39478
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